nuclearnorm

Nuclear norm is a concept in linear algebra and optimization, particularly relevant in the field of matrix analysis. It is defined as the sum of the singular values of a matrix. The singular values of a matrix are the square roots of the eigenvalues of the matrix's transpose times the matrix itself. The nuclear norm is also known as the trace norm or Ky Fan norm.

The nuclear norm is a convex function, which makes it useful in optimization problems where convexity is desirable. It is often used as a regularization term in machine learning algorithms, such as matrix completion and robust principal component analysis. In these applications, the nuclear norm is used to promote low-rank solutions, which can help to reduce overfitting and improve generalization.

The nuclear norm is closely related to the Frobenius norm, which is the sum of the squares of the singular values. While the Frobenius norm is also convex, it is not as effective as the nuclear norm in promoting low-rank solutions. The nuclear norm is also related to the rank of a matrix, which is the number of non-zero singular values. However, the rank is not a convex function, which makes it less suitable for use in optimization problems.

In summary, the nuclear norm is a convex function that is used to promote low-rank solutions in optimization problems. It is defined as the sum of the singular values of a matrix and is closely related to the Frobenius norm and the rank of a matrix.